Inspect the ceiling joists from the attic or crawl space. If the joists are directly above the current ceiling, you will not be able to raise it.
Prepare the room for the removal process. Lay drop cloths down over the floor and tape them into place. Tape drop cloths over the doorways and windows to the room and make sure all of the fixtures and vent are taped off.
Remove the drywall on the ceiling. Insert a pry bar into the seam between the ceiling and the wall and pry down to pop the first drywall panel off of its nails or screws. Keep garbage bins handy to dispose of the panels as you work. Ask a friend to help as these panels can weigh in excess of 40 pounds a piece. If you have a stubborn section of drywall that will not come off, break it apart with a hammer and pull it down by hand.
Remove any remaining nails or screws in the wood framing. Use the back end of a hammer or a nail bar to remove nails. For screws, use a cordless screwdriver set to the reverse position. It helps to have a small bucket handy to dispose of the nails or screws as you go.
Remove the dropped wood frame. If the ceiling was dropped, there will be a wood frame suspended from the joists. Once the drywall is removed, this wood frame can be dismantled with a pry bar and a hammer. You can now install a new ceiling at a higher level or keep it open for the exposed-beam look.